Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [probability-distributions]

Questions on using, finding, or otherwise relating to probability distributions, probability density functions (pdfs), cumulative distribution functions (cdfs), or other related functions. Use this tag along with the tags (probability), (probability-theory) or (statistics).

Any probability distribution, including beta, binomial, chi, Erlang, gamma, geometric, lognormal, negative binomial, normal (Gaussian), Pareto, Poisson, Student's t, uniform, Wald, Weibull, zeta, and Zipf.

28080 questions
0
votes
1 answer

obtaining expected value for moment generating function

This is the given cumulative distribution function: $F(x) = \begin{cases} 0 & x<0\\ 0.5x & 0≤x<1 \\ 0.25(x-1) + 0.5 & 1≤x<3\\ 1 & y≥3 \end{cases}$ We are asked to determine E($X$), E($X^2$), Var($X$), the mgf of X, and also use the mgf to verify…
Denson
  • 91
0
votes
1 answer

What does the Squared 2-Parameter Exponential Cumulative Density Function Measure?

The 2-parameter exponential cumulative density function is defined as $1-e^{\lambda (x-\gamma)}\quad$ (see e.g. this page ) Question: what does $\quad1-\left(1-e^{\lambda (x-\gamma)}\right)^2\quad$ measure? Any information about that…
Manfred Weis
  • 221
0
votes
0 answers

How can I show the accuracy of generated samples from a distribution?

Considering a specific PDf, how can I claim that generated samples are based on the PDF. Is it enough to compare the frequency of generated samples with the PDF curve and to show the mean of generated samples are the same as the mean of PDF?
Han
  • 23
0
votes
1 answer

Moments of the multivariate hypergeometric distribution

Assume the Wikipedia definitions for the multivariate hypergeometric distribution, i.e. let $(X_1,\dots X_c)$ have multivariate hypergeometric distribution with $c\in\mathbb{N}$, $(K_1,\dots,K_c)\in\mathbb{N}^c$, $N=\sum_{i=1}^c K_i$ and…
phinz
  • 139
0
votes
1 answer

Expectation of a uniform distribution

$X$ is uniform over $(0,1)$. What is $E[X|X<\frac12]$? Here's what I did so far, but I'm not sure it's right: $f_X(x|X<1/2)=2$, which is also uniform, so the expected value is just $\frac{a+b}2=\frac{0+\frac12}2=\frac14$
hello888
  • 173
0
votes
1 answer

If we know the mean value of $x$, what is the mean value of the inverse $\frac{1}{x}$?

Random variable $x$ follows normal distribution and we know the mean value of $x$. Is there a well-known way to compute the mean value of $\frac{1}{x}$? Perhaps, some way to estimate the integral of $pdf(x)*(\frac{1}{x})$ from zero to infinity?
apoldem
  • 1
0
votes
1 answer

Probability: Poisson distribution

Molecules are emitted in a rate which distribute poisson(0.5). (0.5 molecule per/second rate). calculate the probabilities: At least one molecule will be emitted in a certain second. More than 3 molecules will be emitted within 5 seconds. thanks.
adamco
  • 413
0
votes
1 answer

1-sigma errors from a non-Gaussian probability distribution

I have an integral-normalised probability distribution $P_x$ that looks like this: The peak of this distribution occurs at $x\approx5.2$. I am trying to find a way to measure the uncertainty on this peak value. Clearly the distribution is not…
ajrlewis
  • 35
0
votes
0 answers

If $X=f(Z)$ and $Y=g(Z)$, what is the joint PDF of $X$ and $Y$?

If two random variables $X$ and $Y$ are functions of the same one random variable $Z$, how do we find the joint PDF of $X$ and $Y$? I have seen examples where two RVs are functions of the other two RVs, but what if they are function of the same RV?…
Confounded
  • 583
0
votes
1 answer

Poisson and Exponential connection

Assume you have two Geiger counters. Time gaps between following molecules that arriving to the first counter= (Ti-Ti-1) ~exp(λ1) Time gaps between following molecules that arriving to the second counter= (Tj-Tj-1) ~exp(λ2) Now assume that over one…
adamco
  • 413
0
votes
1 answer

Distribution convergence, continuous functions with bounded support

I'm stuck with the next problem Let $\mu_n,\mu$ be probability measures; show that if $$\int f\,d\mu_n\to\int f \,d\mu $$ for all continuous $f$ with bounded support, then $\mu_n\to \mu.$ $\mu_n\to\mu$ means $\mu_n(-\infty,t]\to\mu(-\infty,t]$ for…
juaaan
  • 77
0
votes
1 answer

Subtracting a half-normal distribution from a normal distribution

Suppose I have two independent random variables $X_1$ and $X_2$, where the former is a Gaussian distribution and the latter is a half-Gaussian distribution, both with 0 means. $X_1 \sim \mathcal{N}(0, \sigma_1)$ $X_2 \sim \left \vert \mathcal{N}(0,…
SeunSustio
  • 111
0
votes
3 answers

Find cumulative distribution from density distribution with a constant

The PDF of a random variable $X$ is: $f(x) = \begin{cases} 0 , x<2 \ \text{or} \ x >4 \\ a(x-2)(4-x), 2 \leq x \leq 4 \end{cases} $ Find $a$ and $F_X (x)$. My attempt: By taking $\int_{2}^{4} f_X (x) = 1$, I worked out $a = \frac{3}{4}$. Then $F_x…
dunguyen
  • 71
0
votes
1 answer

Find k value where the function is a pdf

Find k value where the function is a pdf (a) $kx^6(1-x)^4$, for $0 < x < 1, 0$ otherwise (b) $kx^2(4-x)^3$, for $0 < x < 4$. $0$ otherwise my attempt (a) $$\int_{0}^{1} kx^6(1-x)^4 dx$$ Do I just solve this on $[0,1]?$
Bas
  • 795
  • 9
  • 27
0
votes
1 answer

Density of vertically truncated normal distribution

I'm interested in the Gaussian density, truncated vertically at a threshold $\beta>0$. $$ p(x)=\frac{1}{Z} \min\left\{\exp\Big(-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)\Big),\beta\right\} \\ $$ How can I find the normalizing constant $Z$?
danijar
  • 681
1 2 3
…
99 100